Among the nonlinear corrections to the linear Maxwell field, it is
of interest to consider the exponential form of the nonlinear
electrodynamics [1]}. Numerical studies on the properties
of the holographic superconductors in the presence of BI nonlinear
electrodynamics [2]} as well as logarithmic and exponential
nonlinear (EN) electrodynamics have been carried out
[3]}, [2]}. In particular, it was shown that EN
electrodynamics correction makes the condensation harder
[3]}. When the gauge field is the BI one, based on the
SL eigenvalue problem, several properties of holographic \(s\) -wave
superconductors in the background of a Schwarzschild-AdS spacetime
have been investigated in the prob limit [6]}, [7]} and away
from the probe limit [8]}, [9]}. As far as we know the
analytical study on the properties of holographic superconductors
with EN electrodynamics remains to be done. In this paper, we
would like to extend the analytical study on the holographic
\(s\) -wave superconductors by taking the gauge field in the form of
EN electrodynamics and using SL analytical techniques. For
economic reasons, we shall concentrate on the probe limit by
neglecting backreaction of the matter fields on the spacetime
metric. We shall compute the critical temperature and critical
exponent for this kind of holographic superconductor by using the
perturbation method. Then, we will compare our results with the
existing numerical results for holographic superconductors with EN
electrodynamics [3]} as well as the analytical results
derived in [6]} for the holographic superconductors with BI
nonlinear electrodynamics.
